3 Generation of comple nonlinear benchmark functions for optimiation using fu sets and classical test functions 73 INTRODUCTION Optimiation (maimiation or minimiation) of continuous nonlinear functions is an important problem in sciences, mathematics, engineering and economics, due to man real-world problems would be solved in these terms []. However, nonlinear optimiation is usuall a hard problem; their difficulties are related to the compleit of goal function, constrains, topolog of the search region and limitations of the optimiation methodolog. Optimiation field has several tpes of problems, e.g. combinatorial optimiation, integer optimiation, quadratic optimiation, and other; and a universal optimier does not eist. Talking about optimiers, Wolpert and Macread emphasie two points: First, when an optimier is designed for a specific problem, it works better for the problem than other generic optimier. Second, there are not harder problems, but there are different tpes of problems. It means that each tpe of problem has specific characteristics and it agrees to them an optimier could have better performance that another one []. I.e. For developing and comparing new methods, and for solving real world problems efficientl, we will want benchmark functions with the highest number of characteristics available or the most closed characteristic to the real problem. Moreover, scientific communit provides special test to compare methods [3]. There are some well-known test functions [4-7]. The are characteried b their behavior, which is manifested b the features of the generated surface, but regularl these functions are limited to specific characteristics. For eample, the Rosenbrock function has its global optimum inside of a narrow valle, while in the Bukin s F4 function the minimum is located inside a canon. The aim of this paper is to present a new methodolog for generating or specifing comple test functions to prove and tune heuristic optimiation algorithms. To do that, we use fu sets theor to combining well-known to generate new functions. The paper is organied as follows. In Section, we review some aspects of classical test functions. Following, fu set theor is introduced in Section 3. In Section 4, we state the description of the proposed methodolog. Net, some eamples are presented in Section 5. Finall, we conclude in Section 6.. CLASSICAL TEST FUNCTIONS The use of well-known test functions is a common procedure for testing global optimiation algorithms, such that, these functions are standard literature benchmarks [4-7]. Thus, researchers are able to compare their results against other methodologies whose results are validated and accepted b the scientific communit. Usuall, a set of functions is used to evaluate the behavior of a specific algorithm in different conditions and to determine their robustness. A classical benchmark is the Rosenbrock s function (Figure ): (, ) ( ) ( ) = f = + () Rosenbrock Figure. Rosenbrock s test function Revista Ingenierías Universidad de Medellín, vol., No. 9, pp ISSN julio-diciembre de /8 p. Medellín, Colombia

4 74 Edd Mesa; Juan David Velásque; Patricia Jaramillo BukinF The homogeneit: the characteristic of the relationship between variables (separable or no separable). The above features would have a great impact in the behavior of a specific optimiation algorithm. For eample, it is well known that gradient based methods have poor behavior optimiing functions with a huge number of local optimal points, because the are trapped in a local optimum and the are not able to escape out. Figure. Bukin F4 test function. f, =. A standard initial point is = and =.. DeJong bounded the function to the interval defined as to use it in heuristic optimiation [8]. And Yao et al. [9] report and etend this function to n dimension using the borders in [ 3,3] n, which increase the compleit because it is a bigger search region. Other classical benchmark is the Bukin s F4 function (Figure ): It has a global minimum in ( ) ( ) = f, = +. + () Proposed b Bukin [] with a global minimum in f (,) =. The standard initial point = and =, and there are not borders available. Test functions are classified in terms of surface s features and the primitive functions used for its construction (or setting-up). Several features are described as following: The dimensionalit of the functions definition: for a fied number of dimensions or in a general wa. The modalit or ruggedness: quantit of local optimal (maimum and minimum) points. The are classified in unimodal functions with a global minimum and multimodal with several minimums. -. FUZZY SETS AND FUZZY OPERATIONS In classical sets theor, the membership of an element to a set is a binar function returning ero when the element not belongs to the set and one in otherwise. In fu set theor, the discrete set {, } for the range of membership function is changed b the interval [, ]. Zero and one still represents the absolute certaint of that element belongs or not belongs to the set. The intermediate values between ero and one represent uncertaint, such that, partial membership is allowed []. The generalied bell function is a tpical method for specifing the membership function of a fu set []: µ = c + b ( abc ;,, ) a (3) Eamples of the generalied bell function with different values for the parameters a, b and c are shown in Figure 3. One-dimensional fu sets, as in equation (3), can be etended to the X Y plane, using a fu set for each ais, and then appling an fu operator. This process is the so-called composition of fu sets e.g. several bi-dimensional fu sets would be obtained using the one-dimensional fu membership functions µ(;, 3, ) and µ(;, 3, ) for representing a fu set for each ais, Universidad de Medellín

6 76 Edd Mesa; Juan David Velásque; Patricia Jaramillo minimums, topolog, etc. Also, this methodolog provides a large number of functions with different characteristics. So, we can obtain a nearer simulation of the method s performance for a specific sort of problems. 4. EXAMPLES To demonstrate the proposed methodolog, we create three new test function combining Rosenbrock s (see Equation ()) and Bukin s F4 (see Equation ()) test functions using the operators described. In the sake of a clear vision of the functions details, we limited the plots in a range in [ 3,3]. In Figure 6, we present the surface obtained when µ (, ) is defined as in equation (4); the obtained function shows that the function BukinF4 s canon is etended with the features of the Rosenbrock s function, while the edges retain the regularit of the function Bukin F4, but with the softness of the Rosenbrock s function. It would be the first function. The generated second function is shown in Figure 7. For this, µ (, ) is defined as in equation (5), as in the previous figure. It still has Composition using product a combination of both functions that produce a different surface, but reminds the same minimum value: min f (, ) = f(, ) =. Moreover, comple surfaces ma be generated using onl one function too. For eample, we use a rotated version of the original Rosenbrock s function as F (, ) in Eq. (6). Figure 8 presents the obtained surface using onl the Rosenbrock function and the product operator. The gradient of the generated function is plot in Figure 9. One of the criteria of compleit for optimiation problems is the eistence of planar regions on the gradient plot, which is evident in the figure. 5. CONCLUSIONS In this paper, we present a novel methodolog to generate new comple test functions for testing, comparing and tuning global optimiation algorithms. Our methodolog is based in the combination of two two-dimensional classical test functions using a weighted sum. These weights are generated b means of a two-dimensional fu set. Thus, using onl two initial test functions, Composition using minimum Figure 6. Composition using the product operator and the Rosenbrock and BukinF4 test functions. Figure 7. Composition using the product operator and the Rosenbrock and Bukin s F4 test functions. Universidad de Medellín

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